{VERSION 2 3 "IBM INTEL NT" "2.3" } {USTYLETAB {CSTYLE "Maple Input" -1 0 "Courier" 0 1 255 0 0 1 0 1 0 0 1 0 0 0 0 }{CSTYLE "2D Math" -1 2 "Times" 0 1 0 0 0 0 0 0 2 0 0 0 0 0 0 }{CSTYLE "Hyperlink" -1 17 "" 0 1 0 128 128 1 0 0 1 0 0 0 0 0 0 } {CSTYLE "2D Input" 2 19 "" 0 1 255 0 0 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "2 D Output" 2 20 "" 0 1 0 0 255 1 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 23 "C ourier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "Help Heading" -1 26 " " 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 256 "" 1 24 0 0 0 0 0 0 0 0 0 0 0 0 0 }{CSTYLE "" -1 257 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 } {CSTYLE "" -1 258 "Courier" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE " " -1 259 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" 23 260 " " 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 261 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" 23 262 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 263 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 } {CSTYLE "" 23 264 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 265 "Courier" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 23 266 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 267 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" 23 268 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 23 269 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 270 "Courier" 1 10 0 0 0 0 0 0 0 0 0 0 3 0 0 }{CSTYLE "" -1 271 "" 0 1 0 0 0 0 1 1 0 0 0 0 0 0 0 }{CSTYLE "" 23 272 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 23 273 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 } {CSTYLE "" 23 274 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 23 275 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 23 276 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" 23 277 "" 0 1 255 0 0 1 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 278 "" 1 14 0 0 0 0 0 1 0 0 0 0 0 0 0 }{CSTYLE "" -1 279 "" 0 1 0 0 0 0 0 1 0 0 0 0 0 0 0 }{PSTYLE "Normal" -1 0 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading 1" 0 3 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 0 0 0 0 0 0 0 }1 0 0 0 6 6 0 0 0 0 0 0 -1 0 }{PSTYLE "Heading \+ 2" 3 4 1 {CSTYLE "" -1 -1 "" 1 14 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 4 4 0 0 0 0 0 0 -1 0 }{PSTYLE "Maple Output" 0 11 1 {CSTYLE "" -1 -1 " " 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 3 0 -1 -1 -1 0 0 0 0 0 0 -1 0 } {PSTYLE "" 11 12 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 } 1 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "Bullet Item" 0 15 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }0 0 0 -1 3 3 0 0 0 0 0 0 15 2 }{PSTYLE "Title" 0 18 1 {CSTYLE "" -1 -1 "" 1 18 0 0 0 0 0 1 1 0 0 0 0 0 0 }3 0 0 -1 12 12 0 0 0 0 0 0 19 0 }{PSTYLE "Author" 0 19 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 }3 0 0 -1 8 8 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 256 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 257 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 258 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 259 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 260 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 261 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }{PSTYLE "" 0 262 1 {CSTYLE "" -1 -1 "" 0 1 0 0 0 0 2 0 0 0 0 0 0 0 0 }0 0 0 -1 -1 -1 0 0 0 0 0 0 -1 0 }} {SECT 0 {EXCHG {PARA 18 "" 0 "" {TEXT 256 32 "Elementary Matrices and \+ Inverses" }}{PARA 19 "" 0 "" {TEXT 257 14 "Brian E. Blank" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 " Keywords" }}{PARA 15 "" 0 "" {HYPERLNK 17 "alias" 2 "alias" "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "backsub" 2 "l inalg[backsub]" "" }{TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "aug ment" 2 "linalg[augment]" "" }{TEXT -1 5 " " }}{PARA 15 "" 0 "" {HYPERLNK 17 "delcols" 2 "linalg[delcols]" "" }{TEXT -1 5 " " }} {PARA 15 "" 0 "" {HYPERLNK 17 "forwardsub" 2 "linalg[forwardsub]" "" } {TEXT -1 0 "" }}{PARA 15 "" 0 "" {HYPERLNK 17 "inverse" 2 "linalg[inve rse]" "" }{TEXT -1 5 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 9 " Code for" }{MPLTEXT 1 0 18 " elementary Matrix " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 8 "restart;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 926 "elementaryMatrix := proc()\nlocal n, i, j, Id;\n\nif nargs < 3 or nargs > 4 then\nERROR(`elementaryMatrix expects three or four argumen ts.`);\nfi:\n\nn := args[1]:\n\nif not type(n,posint) then ERROR(`elem entaryMatrix expects its first argument to be a positive integer.`);\n fi:\n\n\nId := array(1..n,1..n,identity);\ni := args[2]:\nj := args[3] :\n\nif not type(i,posint) or not type(j, posint) then\n ERROR(`elemen taryMatrix expects its second and third arguments to be positive integ ers.`);\nelif i > n or j > n then\nERROR(`elementaryMatrix expects its second and third arguments to be positive integers not greater than t he first argument.`);\nfi:\n\n\n\nif nargs = 3 then\n if i = j then \+ \n ERROR(`When called with three arguments elementaryMatrix expects it s last two arguments to differ.`);\n\nelse\nRETURN(linalg[swaprow](Id, i,j));\nfi:\n\n\n\nelif i = j then\nRETURN(linalg[mulrow](Id,i,args[4] ));\n\nelse\nRETURN(linalg[addrow](Id,i,j,args[4]));\nfi:\nend;" }} {PARA 12 "" 0 "" {TEXT -1 0 "" }}{PARA 12 "" 1 "" {XPPMATH 20 "6#>%1el ementaryMatrixG:6\"6&%\"nG%\"iG%\"jG%#IdGF&F&C*@$529#\"\"$2\"\"%F0-%&E RRORG6#%RelementaryMatrix~expects~three~or~four~arguments.G>8$&9\"6#\" \"\"@$4-%%typeG6$F9%'posintG-F56#%`oelementaryMatrix~expects~its~first ~argument~to~be~a~positive~integer.G>8'-%&arrayG6%;F=F9FL%)identityG>8 %&F;6#\"\"#>8&&F;6#F1@&43-FA6$FOFC-FA6$FTFC-F56#%[pelementaryMatrix~ex pects~its~second~and~third~arguments~to~be~positive~integers.G52F9FO2F 9FT-F56#%_relementaryMatrix~expects~its~second~and~third~arguments~to~ be~positive~integers~not~greater~than~the~first~argument.G@'/F0F1@%/FO FT-F56#%fpWhen~called~with~three~arguments~elementaryMatrix~expects~it s~last~two~arguments~to~differ.G-%'RETURNG6#-&%'linalgG6#%(swaprowG6%F HFOFTFdo-Fio6#-&F]p6#%'mulrowG6%FHFO&F;6#F3-Fio6#-&F]p6#%'addrowG6&FHF OFTFhpF&F&" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 15 " Description of" }{MPLTEXT 1 0 17 " elementaryMatrix" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT 26 9 "F unction:" }{MPLTEXT 1 0 17 " elementaryMatrix" }{TEXT -1 31 " - create s an elementary matrix" }}{PARA 0 "" 0 "usage" {TEXT 26 18 "\nCalling \+ Sequence:" }{TEXT -1 4 "\n " }{TEXT 258 1 "e" }{MPLTEXT 1 0 15 "leme ntaryMatrix" }{TEXT 265 14 "(n, r1, r2, c)" }}{PARA 0 "" 0 "" {TEXT 26 12 "\nParameters:" }{TEXT -1 4 "\n " }{TEXT 264 1 "n" }{TEXT 263 4 " - " }{TEXT -1 22 "a positive integer\n " }{TEXT 262 2 "r1" } {TEXT 261 39 " - a positive integer not greater than " }{TEXT 266 1 "n " }{TEXT 267 1 " " }{TEXT -1 4 "\n " }{TEXT 260 2 "r2" }{TEXT 259 39 " - a positive integer not greater than " }{TEXT 268 1 "n" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }{TEXT 269 1 "c" }{TEXT 270 4 " - " } {TEXT -1 23 "an algebraic expression" }}}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 14 "Description of" }{MPLTEXT 1 0 16 " transformPlane\n" }}{PARA 15 " " 0 "" {TEXT -1 10 "The calls " }{MPLTEXT 1 0 26 " elementaryMatrix(n, r1,r2)" }{TEXT -1 6 " and " }{MPLTEXT 1 0 28 " elementaryMatrix(n,r1, r2,c)" }{TEXT -1 35 " return an elementary matrix with " }{MPLTEXT 1 0 1 "n" }{TEXT -1 19 " rows and columns. " }}{PARA 15 "" 0 "" {TEXT -1 9 "The call " }{MPLTEXT 1 0 23 "elementaryMatrix(n,i,j)" }{TEXT -1 57 " returns the elementary matrix obtained by swapping rows " } {MPLTEXT 1 0 1 "i" }{TEXT -1 5 " and " }{MPLTEXT 1 0 1 "j" }{TEXT -1 8 " of the " }{MPLTEXT 1 0 3 "nxn" }{TEXT -1 17 " identity matrix." }} {PARA 15 "" 0 "" {TEXT -1 9 "The call " }{MPLTEXT 1 0 25 "elementaryMa trix(n,i,i,c)" }{TEXT -1 61 " returns the elementary matrix obtained b y multiplying row " }{MPLTEXT 1 0 1 "i" }{TEXT -1 8 " of the " } {MPLTEXT 1 0 3 "nxn" }{TEXT -1 20 " identity matrix by " }{MPLTEXT 1 0 1 "c" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 9 "The call " } {MPLTEXT 1 0 25 "elementaryMatrix(n,i,j,c)" }{TEXT -1 57 " returns the elementary matrix obtained by adding to row " }{MPLTEXT 1 0 1 "i" } {TEXT -1 9 " of the " }{MPLTEXT 1 0 3 "nxn" }{TEXT -1 18 " identity m atrix " }{MPLTEXT 1 0 1 "c" }{TEXT -1 12 " times row " }{MPLTEXT 1 0 1 "j" }{TEXT -1 1 "." }}{PARA 15 "" 0 "" {TEXT -1 10 "Load the " } {MPLTEXT 1 0 7 "linalg " }{TEXT -1 24 "package prior to using " } {MPLTEXT 1 0 16 "elementaryMatrix" }{TEXT -1 2 ". " }}{PARA 15 "" 0 " " {TEXT -1 9 "Because " }{MPLTEXT 1 0 16 "elementaryMatrix" }{TEXT -1 74 " is not a builtin function its code must be executed prior to \+ calling it." }}{PARA 15 "" 0 "" {TEXT -1 9 "Because " }{MPLTEXT 1 0 16 "elementaryMatrix" }{TEXT -1 92 " has a long name it may be desira ble to use an alias for it. See the examples for this use." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{SECT 0 {PARA 4 "" 0 "" {TEXT -1 9 "Examples:" }}{PARA 0 "" 0 "" {TEXT -1 0 " " }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "elementaryMatrix(5,1,4); #Three argument row swap call" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'M ATRIXG6#7'7'\"\"!F(F(\"\"\"F(7'F(F)F(F(F(7'F(F(F)F(F(7'F)F(F(F(F(7'F(F (F(F(F)" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 50 "elementaryMatrix (6,1,2,0); #Create Identity matrix" }}{PARA 11 "" 1 "" {XPPMATH 20 "6# -%'MATRIXG6#7(7(\"\"\"\"\"!F)F)F)F)7(F)F(F)F)F)F)7(F)F)F(F)F)F)7(F)F)F )F(F)F)7(F)F)F)F)F(F)7(F)F)F)F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 54 "elementaryMatrix(5,3,3,7); #Multiply a row by a numbe r" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)F) 7'F)F(F)F)F)7'F)F)\"\"(F)F)7'F)F)F)F(F)7'F)F)F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 65 "elementaryMatrix(5,2,2,x+x^2/3); #Multipl y a row by an expression" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG 6#7'7'\"\"\"\"\"!F)F)F)7'F),&%\"xGF(*$F,\"\"##F(\"\"$F)F)F)7'F)F)F(F)F )7'F)F)F)F(F)7'F)F)F)F)F(" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 62 "elementaryMatrix(5,4,3,7); #Add a multiple of a row to another" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)F)7'F)F (F)F)F)7'F)F)F(\"\"(F)7'F)F)F)F(F)7'F)F)F)F)F(" }}}{EXCHG {PARA 0 "> \+ " 0 "" {MPLTEXT 1 0 59 "alias(E = elementaryMatrix); #Using an \"alia s\" or nickname" }}{PARA 11 "" 1 "" {XPPMATH 20 "6$%\"IG%\"EG" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "E(5,4,3,7);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)F)7'F)F(F)F)F)7'F)F )F(\"\"(F)7'F)F)F)F(F)7'F)F)F)F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 32 " Inverses of Elementary Matrices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT 271 93 "Every elementary matrix is invertible. I t's inverse is an elementary matrix of the same type." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 17 "The inverse of " }{MPLTEXT 1 0 8 "E(n,i,j)" }{TEXT -1 13 " is itself." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "E(5,3,4) &* \+ E(5,3,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$-%'MATRIXG6#7'7' \"\"\"\"\"!F,F,F,7'F,F+F,F,F,7'F,F,F,F+F,7'F,F,F+F,F,7'F,F,F,F,F+F&" } }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\" \"\"\"\"!F)F)F)7'F)F(F)F)F)7'F)F)F(F)F)7'F)F)F)F(F)7'F)F)F)F)F(" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The inverse of " }{MPLTEXT 1 0 10 "E(n,i,i,c)" } {TEXT -1 7 " is " }{MPLTEXT 1 0 12 "E(n,i,i,1/c)" }{TEXT -1 1 "." } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 27 "E(5,2,2,c) &* E(5,2,2,1/c); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$-%'MATRIXG6#7'7'\"\"\"\"\" !F,F,F,7'F,%\"cGF,F,F,7'F,F,F+F,F,7'F,F,F,F+F,7'F,F,F,F,F+-F'6#7'F*7'F ,*$F.!\"\"F,F,F,F/F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "ev alm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\" !F)F)F)7'F)F(F)F)F)7'F)F)F(F)F)7'F)F)F)F(F)7'F)F)F)F)F(" }}}{PARA 0 " " 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 " " {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 16 "The inverse of " } {MPLTEXT 1 0 10 "E(n,i,j,c)" }{TEXT -1 7 " is " }{MPLTEXT 1 0 11 "E (n,i,j,-c)" }{TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "E(5,4,3,7) &* E(5,4,3,-7);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$-%'MATRIXG6#7'7'\"\"\"\"\"!F,F ,F,7'F,F+F,F,F,7'F,F,F+\"\"(F,7'F,F,F,F+F,7'F,F,F,F,F+-F'6#7'F*F-7'F,F ,F+!\"(F,F0F1" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\"); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7'7'\"\"\"\"\"!F)F)F)7 'F)F(F)F)F)7'F)F)F(F)F)7'F)F)F)F(F)7'F)F)F)F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 38 " Multiplication \+ by Elementary Matrices" }}{PARA 0 "" 0 "" {TEXT -1 32 "When we left-m ultiply a matrix " }{TEXT 272 3 " A " }{TEXT -1 26 " by an elementary \+ matrix " }{TEXT 273 1 "E" }{TEXT -1 62 " the product is the matrix t hat is obtained by performing on " }{TEXT 274 3 " A " }{TEXT -1 46 " t he elementary row operation that generates " }{TEXT 275 1 "E" }{TEXT -1 28 " from the identity matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 58 "A := matrix(4,4,[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$ \"\"%7&\"\"&\"\"'\"\"(\"\")7&\"\"*\"#5\"#6\"#77&\"#8\"#9\"#:\"#;" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 15 " E(4,1,4) &* A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$-%'MATRIXG6#7&7&\"\"!F+F+\"\"\"7&F+F,F+ F+7&F+F+F,F+7&F,F+F+F+%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"#8 \"#9\"#:\"#;7&\"\"&\"\"'\"\"(\"\")7&\"\"*\"#5\"#6\"#77&\"\"\"\"\"#\"\" $\"\"%" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "E(4,2,2,c) &* A; " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$-%'MATRIXG6#7&7&\"\"\"\"\" !F,F,7&F,%\"cGF,F,7&F,F,F+F,7&F,F,F,F+%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MA TRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"%7&,$%\"cG\"\"&,$F.\"\"',$F.\"\"(,$F.\" \")7&\"\"*\"#5\"#6\"#77&\"#8\"#9\"#:\"#;" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "E(4,2,3,c) &* A;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#- %#&*G6$-%'MATRIXG6#7&7&\"\"\"\"\"!F,F,7&F,F+F,F,7&F,%\"cGF+F,7&F,F,F,F +%\"AG" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "evalm(\");" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"#\"\"$\"\"%7 &\"\"&\"\"'\"\"(\"\")7&,&\"\"*F(%\"cGF-,&\"#5F(F4F.,&\"#6F(F4F/,&\"#7F (F4F07&\"#8\"#9\"#:\"#;" }}}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT -1 0 "" }}}{SECT 1 {PARA 3 "" 0 "" {TEXT -1 56 " Invertib le Matrices are Products of Elementary Matrices" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 213 "We can row reduce an inv ertible matrix to the identity matrix. If the row reduction operations needed to do that are implemented as left multiplications by the corr esponding elementary matrices then this says that" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 19 1 "E[s ]*E[s-1]*`...`*E[2]*E[1]*A=I" "/*.&%\"EG6#%\"sG\"\"\"&F%6#,&F'F(F(!\" \"F(%$...GF(&F%6#\"\"#F(&F%6#F(F(%\"AGF(%\"IG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " or" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 19 1 "A=( E[1])^(-1)*E[2]^(-1)*`...`*E[s-1]^(-1)*E[s]^(-1)" "/%\"AG*,)&%\"EG6#\" \"\",$F)!\"\"F))&F'6#\"\"#,$F)F+F)%$...GF))&F'6#,&%\"sGF)F)F+,$F)F+F)) &F'6#F6,$F)F+F)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 85 " But the inverse of an elementary matrix is an elementary matrix.\n\nNotice that " }{XPPEDIT 19 1 "A^(-1)=E[s]*E[s-1]*`.. .`*E[2]*E[1]" "/)%\"AG,$\"\"\"!\"\"*,&%\"EG6#%\"sGF&&F*6#,&F,F&F&F'F&% $...GF&&F*6#\"\"#F&&F*6#F&F&" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 57 "In other words, when we row reduce an invertible matrix " }{TEXT 276 1 "A" }{TEXT -1 109 " to the identity matrix, the same row operations applied to t he identity matrix results in the inverse of " }{TEXT 277 1 "A" } {TEXT -1 1 "." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT 278 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 57 "A := matrix([[1,2,0,-1],[1,1,1,0],[0,0,1, 1],[-1,2,0,0]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6# 7&7&\"\"\"\"\"#\"\"!!\"\"7&F*F*F*F,7&F,F,F*F*7&F-F+F,F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B := E(4,1,2,0);" }}{PARA 11 "" 1 " " {XPPMATH 20 "6#>%\"BG-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+ F*F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "C := augm ent(A,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIXG6#7&7*\" \"\"\"\"#\"\"!!\"\"F*F,F,F,7*F*F*F*F,F,F*F,F,7*F,F,F*F*F,F,F*F,7*F-F+F ,F,F,F,F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "C1 := addrow (C,1,4,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G-%'MATRIXG6#7&7*\" \"\"\"\"#\"\"!!\"\"F*F,F,F,7*F*F*F*F,F,F*F,F,7*F,F,F*F*F,F,F*F,7*F,\" \"%F,F-F*F,F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E1 := E( 4,1,4,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G-%'MATRIXG6#7&7&\" \"\"\"\"!F+F+7&F+F*F+F+7&F+F+F*F+7&F*F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "C2 := addrow(C1,1,2,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G-%'MATRIXG6#7&7*\"\"\"\"\"#\"\"!!\"\"F*F,F,F,7*F, F-F*F*F-F*F,F,7*F,F,F*F*F,F,F*F,7*F,\"\"%F,F-F*F,F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E2 := E(4,1,2,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7&!\"\"F*F+F+ 7&F+F+F*F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C3 \+ := addrow(C2,2,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C3G-%'MATRI XG6#7&7*\"\"\"\"\"!\"\"#F*!\"\"F,F+F+7*F+F-F*F*F-F*F+F+7*F+F+F*F*F+F+F *F+7*F+\"\"%F+F-F*F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E3 := E(4,2,1,2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G-%'MATRIXG 6#7&7&\"\"\"\"\"#\"\"!F,7&F,F*F,F,7&F,F,F*F,7&F,F,F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C4 := addrow(C3,2,4,4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C4G-%'MATRIXG6#7&7*\"\"\"\"\"!\"\"#F*!\" \"F,F+F+7*F+F-F*F*F-F*F+F+7*F+F+F*F*F+F+F*F+7*F+F+\"\"%\"\"$!\"$F1F+F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E4 := E(4,2,4,4);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E4G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7 &F+F*F+F+7&F+F+F*F+7&F+\"\"%F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 22 "C5 := mulrow(C4,2,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C5G-%'MATRIXG6#7&7*\"\"\"\"\"!\"\"#F*!\"\"F,F+F+7*F+F*F-F-F*F -F+F+7*F+F+F*F*F+F+F*F+7*F+F+\"\"%\"\"$!\"$F1F+F*" }}}{EXCHG {PARA 0 " > " 0 "" {MPLTEXT 1 0 18 "E5 := E(4,2,2,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E5G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7&F+!\"\"F+F+7&F+F +F*F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "C6 := ad drow(C5,3,1,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C6G-%'MATRIXG6# 7&7*\"\"\"\"\"!F+!\"\"F,\"\"#!\"#F+7*F+F*F,F,F*F,F+F+7*F+F+F*F*F+F+F*F +7*F+F+\"\"%\"\"$!\"$F2F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E6 := E(4,3,1,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E6G-%'MAT RIXG6#7&7&\"\"\"\"\"!!\"#F+7&F+F*F+F+7&F+F+F*F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C7 := addrow(C6,3,2,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C7G-%'MATRIXG6#7&7*\"\"\"\"\"!F+!\"\"F,\" \"#!\"#F+7*F+F*F+F+F*F,F*F+7*F+F+F*F*F+F+F*F+7*F+F+\"\"%\"\"$!\"$F2F+F *" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E7 := E(4,3,2,1);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E7G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7 &F+F*F*F+7&F+F+F*F+7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "C8 := addrow(C7,3,4,-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C 8G-%'MATRIXG6#7&7*\"\"\"\"\"!F+!\"\"F,\"\"#!\"#F+7*F+F*F+F+F*F,F*F+7*F +F+F*F*F+F+F*F+7*F+F+F+F,!\"$\"\"%!\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E8 := E(4,3,4,-4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%#E8G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7&F+F+F*F+7&F+F+!\"%F* " }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C9 := addrow(C8,4,3,1); " }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C9G-%'MATRIXG6#7&7*\"\"\"\"\"!F +!\"\"F,\"\"#!\"#F+7*F+F*F+F+F*F,F*F+7*F+F+F*F+!\"$\"\"%F1F*7*F+F+F+F, F1F2!\"%F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "E9 := E(4,4,3 ,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E9G-%'MATRIXG6#7&7&\"\"\"\" \"!F+F+7&F+F*F+F+7&F+F+F*F*7&F+F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C10 := mulrow(C9,4,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$C10G-%'MATRIXG6#7&7*\"\"\"\"\"!F+!\"\"F,\"\"#!\"#F+7*F+F*F+F+ F*F,F*F+7*F+F+F*F+!\"$\"\"%F1F*7*F+F+F+F*\"\"$!\"%F2F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 19 "E10 := E(4,4,4,-1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$E10G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F+7&F+F*F+F+7& F+F+F*F+7&F+F+F+!\"\"" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 25 "C1 1 := addrow(C10,4,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%$C11G-%'M ATRIXG6#7&7*\"\"\"\"\"!F+F+\"\"#!\"#F,!\"\"7*F+F*F+F+F*F.F*F+7*F+F+F*F +!\"$\"\"%F1F*7*F+F+F+F*\"\"$!\"%F2F." }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E11 := E(4,4,1,1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6 #>%$E11G-%'MATRIXG6#7&7&\"\"\"\"\"!F+F*7&F+F*F+F+7&F+F+F*F+7&F+F+F+F* " }}}{PARA 0 "" 0 "" {TEXT -1 2 "\n\n" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 31 "A_inverse := delcols(C11,1..4);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%*A_inverseG-%'MATRIXG6#7&7&\"\"#!\"#F*!\"\"7&\"\"\"F, F.\"\"!7&!\"$\"\"%F1F.7&\"\"$!\"%F2F," }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "evalm(A &* A_inverse); #Verification!" }}{PARA 11 " " 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"\"\"\"!F)F)7&F)F(F)F)7&F)F)F (F)7&F)F)F)F(" }}}{PARA 0 "" 0 "" {TEXT -1 89 "\nAnother way to get th e inverse is to multiply the elementary matrices we have generated:" } }{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 45 "E11&*E10&*E9&*E8&*E7&*E6&*E5 &*E4&*E3&*E2&*E1;" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%#&*G6$-F$6$-F$6 $-F$6$-F$6$-F$6$-F$6$-F$6$-F$6$-F$6$%$E11G%$E10G%#E9G%#E8G%#E7G%#E6G%# E5G%#E4G%#E3G%#E2G%#E1G" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 9 "e valm(\");" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7&7&\"\"#!\"# F(!\"\"7&\"\"\"F*F,\"\"!7&!\"$\"\"%F/F,7&\"\"$!\"%F0F*" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}{PARA 0 "" 0 "" {TEXT -1 64 "Of course there is a simple command for the inverse of a matrix:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 11 "inverse(A);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%' MATRIXG6#7&7&\"\"#!\"#F(!\"\"7&\"\"\"F*F,\"\"!7&!\"$\"\"%F/F,7&\"\"$! \"%F0F*" }}}{PARA 0 "" 0 "" {TEXT -1 1 "\n" }}}{SECT 1 {PARA 3 "" 0 " " {TEXT -1 17 " LU Decomposition" }}{PARA 0 "" 0 "" {TEXT -1 1 " " }} {PARA 0 "" 0 "" {TEXT -1 58 " If a matrix can be row reduced to upper \+ triangular form " }{XPPEDIT 19 1 "U" "I\"UG6\"" }{TEXT -1 89 " using only the row operation of adding a multiple of one row to a lower, \+ then we have" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 5 " " }{XPPEDIT 19 1 "E[t]*E[t-1]*`...`*E[2]*E[1]*A=U" "/*.&%\" EG6#%\"tG\"\"\"&F%6#,&F'F(F(!\"\"F(%$...GF(&F%6#\"\"#F(&F%6#F(F(%\"AGF (%\"UG" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 13 " where each " }{XPPEDIT 19 1 "E[i]" "&%\"EG6#%\"iG" }{TEXT -1 100 " \+ is lower triangular. But we have seen that the inverse of is the sam e type of matrix. Therefore " }{XPPEDIT 19 1 "E[i]^(-1)" ")&%\"EG6#% \"iG,$\"\"\"!\"\"" }{TEXT -1 41 " is also lower triangular and there fore" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 4 " \+ " }{XPPEDIT 19 1 "A=E[1]^(-1)*E[2]^(-1)*`...`*E[t]^(-1)*U" "/%\"AG*, )&%\"EG6#\"\"\",$F)!\"\"F))&F'6#\"\"#,$F)F+F)%$...GF))&F'6#%\"tG,$F)F+ F)%\"UGF)" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 74 "expresses as a lower triangular matrix times an upper triangular \+ matrix " }{XPPEDIT 19 1 "A=L*U" "/%\"AG*&%\"LG\"\"\"%\"UGF&" }{TEXT -1 227 ". The method of finding such a decomposition is quite similar \+ to finding an inverse. However one uses only certain row reductions an d one stops when the row reduction reaches an upper triangular matrix, not the identity matrix." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 " " 0 "" {TEXT 279 8 "Example:" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 39 "A := matrix([[1,2,3],[2,1,2] ,[2,0,1]]);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"AG-%'MATRIXG6#7%7% \"\"\"\"\"#\"\"$7%F+F*F+7%F+\"\"!F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 16 "B := E(3,1,2,0);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#> %\"BG-%'MATRIXG6#7%7%\"\"\"\"\"!F+7%F+F*F+7%F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "C := augment(A,B);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"CG-%'MATRIXG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F+F*F+ F-F*F-7(F+F-F*F-F-F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 23 "C1 \+ := addrow(C,1,2,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C1G-%'MATRI XG6#7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-!\"$!\"%!\"#F*F-7(F+F-F*F-F-F*" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E1 := E(3,1,2,-2);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E1G-%'MATRIXG6#7%7%\"\"\"\"\"!F+7%! \"#F*F+7%F+F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 24 "C2 := ad drow(C1,1,3,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C2G-%'MATRIXG6# 7%7(\"\"\"\"\"#\"\"$F*\"\"!F-7(F-!\"$!\"%!\"#F*F-7(F-F0!\"&F1F-F*" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "E2 := E(3,1,3,-2);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E2G-%'MATRIXG6#7%7%\"\"\"\"\"!F+7%F+F*F+7 %!\"#F+F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 26 "C3 := addrow(C 2,2,3,-4/3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#C3G-%'MATRIXG6#7%7( \"\"\"\"\"#\"\"$F*\"\"!F-7(F-!\"$!\"%!\"#F*F-7(F-F-#F*F,#F+F,#F0F,F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 20 "E3 := E(3,2,3,-4/3);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%#E3G-%'MATRIXG6#7%7%\"\"\"\"\"!F+7%F +F*F+7%F+#!\"%\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 49 "U \+ := delcols(C3, 4..6); L1 := delcols(C3, 1..3);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"UG-%'MATRIXG6#7%7%\"\"\"\"\"#\"\"$7%\"\"!!\"$!\"%7% F.F.#F*F," }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%#L1G-%'MATRIXG6#7%7%\" \"\"\"\"!F+7%!\"#F*F+7%#\"\"#\"\"$#!\"%F1F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 47 "evalm(L1 - E3 &* E2 &* E1); #Verification of L1" } }{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"!F(F(F'F'" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 48 "evalm(L1 &* A - U); #Verific ation that (L1)A = U" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7% 7%\"\"!F(F(F'F'" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 17 "L := inv erse(L1);" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"LG-%'MATRIXG6#7%7%\" \"\"\"\"!F+7%\"\"#F*F+7%F-#\"\"%\"\"$F*" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 44 "evalm(L &* U - A); #Verification that A = LU" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#-%'MATRIXG6#7%7%\"\"!F(F(F'F'" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 45 " Using the LU Decomposition to So lve a System" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 7 "Solve " }{XPPEDIT 19 1 "A * MATRIX([[x], [y], [z]]) = MATRIX([[ 5], [2], [3]] " "/*&%\"AG\"\"\"-%'MATRIXG6#7%7#%\"xG7#%\"yG7#%\"zGF%-F '6#7%7#\"\"&7#\"\"#7#\"\"$" }{TEXT -1 31 " using the LU decompositio n. " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 112 "T o solve Ar = s or (LU)r = s we may find a t such that Lt = \+ s and then an r such that Ur = t. Then" }}{PARA 0 "" 0 "" {TEXT -1 126 "(LU)(r) = L(Ur) = Lt = s. The advantage is that solving systems \+ with upper and lower triangular matrices can be done entirely" }} {PARA 0 "" 0 "" {TEXT -1 45 "by backsubstitution and forward substitut ion." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "s := vector([5,2,3]);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"sG-%'VECTORG6#7%\"\"&\"\"#\"\"$" } }}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 21 "t := forwardsub(L,s);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"tG-%'VECTORG6#7%\"\"&!\")#\"#6\"\" $" }}}{EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 18 "r := backsub(U,t);" }} {PARA 11 "" 1 "" {XPPMATH 20 "6#>%\"rG-%'VECTORG6#7%!\"%!#7\"#6" }}} {EXCHG {PARA 0 "> " 0 "" {MPLTEXT 1 0 29 "linsolve(A,s); #Verificatio n" }}{PARA 11 "" 1 "" {XPPMATH 20 "6#-%'VECTORG6#7%!\"%!#7\"#6" }}} {PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}} {SECT 1 {PARA 3 "" 0 "" {TEXT -1 33 " Copyright and Author Information " }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 256 "" 0 "" {TEXT -1 58 "ele mentaryMatricesR4.mws A MapleV Release 4 worksheet." }}{PARA 257 " " 0 "" {TEXT -1 0 "" }}{PARA 258 "" 0 "" {TEXT -1 44 "Author: Brian E . Blank (19 September 2001)" }}{PARA 259 "" 0 "" {TEXT -1 0 "" }} {PARA 0 "" 0 "" {TEXT -1 51 "This document may not be distributed by a ny medium," }}{PARA 0 "" 0 "" {TEXT -1 55 "including print, disk, and \+ electronic transfer, without" }}{PARA 0 "" 0 "" {TEXT -1 39 "prior wri tten permission of the author." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }} {PARA 260 "" 0 "" {TEXT -1 49 "For more information, please contact t he author:" }}{PARA 261 "" 0 "" {TEXT -1 4 " " }}{PARA 261 "" 0 "" {TEXT -1 32 " Department of Mathematics, " }}{PARA 0 "" 0 "" {TEXT -1 39 " Washington University in St. Louis" }}{PARA 0 "" 0 " " {TEXT -1 26 " St. Louis, MO 63130" }}{PARA 0 "" 0 "" {TEXT -1 3 " " }}{PARA 0 "" 0 "" {TEXT -1 33 " Telephone: (314) 935-67 63" }}{PARA 262 "" 0 "" {TEXT -1 44 " e-mail: brian@mat h.wustl.edu" }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}{PARA 0 "" 0 "" {TEXT -1 61 " Copyright: \251 2001 Brian E. Blank, All Rights Reserved ." }}{PARA 0 "" 0 "" {TEXT -1 0 "" }}}}{MARK "8" 0 }{VIEWOPTS 1 1 0 1 1 1803 }